Create an outline mesh around another structure / edges with equal distance
Considering the base shape has no surface and is edges only, you can convert it to a curve with AltC, then use the "extrude" parameter:



Doing that the distance from the outer edges to the base vertices is constant.

Edit: in meantime I've downloaded your file. From both solution the problem is your shape is scaled. So you need to apply scale CtrlA before any other thing.